RULES OF INFERENCE AND RULES OF REPLACEMENT1

1. Modus Ponens P⊃Q

_P_:. Q

2. Modus Tollens P⊃Q ~ Q_

:. ~P3. Hypothetical Syllogism

P⊃Q Q ⊃ R :. P⊃R

4. Disjunctive Syllogism PVQ

_~P_:. Q

5.Constructive Dilemma (P⊃Q) o (R⊃S) PVR :.QVS

7. Destructive Dilemma (P⊃Q) o (R⊃S) ~ Q V ~ S :. ~P V~R

6. Simplification P o Q :.P :.Q8 Addition P_ :.P V Q

9. Conjunction P Q_ :.P o Q

10. Absorption P ⊃ Q :. P ⊃ ( P o Q)

RULES OF REPLACEMENT: Within a context of a proof, logically equivalent expressions may replace each other

10. De Morgan’s Rule ~(P o Q) ≡ (~PV~Q)~(PVQ) ≡ (~P o ~Q)

11. Commutation (PVQ) ≡ (QVP)(P o Q) ≡ (Q o P)

12. Association [PV (QVR)] ≡ [(PVQ) VR][Po (Q o R)] ≡ [(P o Q) o R]

13. Distribution [Po(QVR)] ≡ [(P o Q) V (P o R)][PV (Q o R)] ≡ [(P V Q) o (PVR)]

14. Double Negation P≡ ~~P15. Transposition (P⊃Q) ≡ (~Q⊃~P)16. Material Implication (P⊃Q) ≡ (~PVQ)17. Material Equivalence (P≡ Q) ≡ [(P ⊃ Q) o (Q ⊃ P)]

(P≡ Q) ≡ [(P o Q) V (~P o ~Q)]18. Exportation [(P o Q)⊃R)] ≡ [P⊃(Q⊃R)]19. Tautology P≡ (PVP)

P≡ (P o P)

11 Condensed by Roland L. Aparece, MA from Dan Magat, A First Book in Logic (Manila: Felta, 1991) p.60 and Patrick Hurley, A Concise Introduction to Logic.(Belmont: Wadswoth/Thompson Learning, 2000) pp. 370,379-380,389-390,399-400.

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STRATEGIES FOR APPLYING THE RULES OF INFERENCE AND RULES OF REPLACEMENT

TO PROVE STRATEGY1. A statement letter or the negation or a sentence letter

Use Modus Ponens, Modus Tollens, Disjunctive Syllogism

2. A o B Work for A and work for B, then use conjunction3. ~(A o B) Work for the equivalent disjunction

~ A V ~B then apply De Morgan’s rule4. A V B Work for A and infer A V B by Addition or use

Constructive Dilemma5. ~A V ~B Work for the equivalent conjunction ~A o ~B then apply

De Morgan’s rule6. A ⊃ B Use any strategy for 1, or conditional proof. Work for the

equivalent ~A V B then apply Material Implication. Sometime Hypothetical Syllogism will do also.

7. ~(A ⊃ B) Work for the equivalent ~(~A V B) then apply Material Implication.

8. A ≡ B Work for A ⊃ B, and B ⊃ A separately and derive the conjunction (A ⊃ B) o (B ⊃ A).

Or work for [(A o B) V (~A ⊃ ~B)] by Disjunctive Syllogism or constructive Dilemma.

Note if A≡B occurs as a premise then the first step which you need is to break or translate the equivalence to(A ⊃ B) o (B ⊃ A), or (A o B) V (~A o ~ B).

9. ~(A ≡ B) Work for (A ⊃ B) ⊃ (B o ~A) using the strategies outlined above for implications and conjunctions.

10. A ⊃ A To derive (A ⊃ A), or any tautology, the best strategy is to use conditional proof or indirect proof)

1. Always begin by attempting to find the conclusion in the premises.

Given~JJ V KK ⊃ LProveL

Let us examine the above argument in detail. The conclusion is L. Upon inspection, we can find K ⊃ L in the premises wherein K is the antecedent of the consequent L. More so, K is found in another premise, a disjunctive statement, J V K. In this case, the “partner” or other disjunct of K is J. Lastly, we have a single letter ~J. To solve this argument, one could simply infer mentally the flow of the solution: J V K and ~J, by applying disjunctive syllogism to the two premises, one could get K. Now we have K ⊃ L and K, by applying Modus Ponens to the two premises, one could get L. Thus, the argument is valid as demonstrated and the key to this solution is by starting to find the conclusion in the premises.

J V K given~J K Disjunctive SyllogismK ⊃ L givenK aboveL Modus Ponens

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Q.E.D.2. If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via Modus Ponens.

GivenA⊃B A⊃B givenA AProve B Modus PonensB Q.E.D

3. If the conclusion contains a negated letter and that appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via Modus Tollens.

GivenC ⊃ B A⊃B A⊃B ~B given~B ~A Modus TollensProve Q.E.D~A

4. If the conclusion is a conditional statement, consider obtaining it via Hypothetical syllogism.GivenA⊃C A⊃C givenC⊃B C⊃BProve A⊃B Hypothetical SyllogismA⊃B

5. If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via Disjunctive Syllogism.

GivenA V B A V B given~A ~AProve B Disjunctive SyllogismB Q.E.D

6. If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining it via simplification.

Given A o B A o B given Prove A Simplification

A Q.E.D.

7. If the conclusion is a conjunctive statement, consider obtaining it via conjunction by first obtaining the individual conjuncts.

Given A ⊃ B A given A C

C A o C Conjunction Prove Q.E.D A o C .

8. If the conclusion is a disjunctive statement, consider obtaining it via Constructive Dilemma or Addition. Given (A⊃B) o (C⊃D) (A⊃B) o (C⊃D) given B⊃C AVC given

AVC BVD Constructive Dilemma Prove Q.E.D

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BVDGivenAVC B givenB BVD AdditionProve Q.E.DBVD

9. If the conclusion contains a letter not found in the premises, Addition must be used to obtain that letter. (See second example under strategy 8.)

10. Conjunction can be used to set up De Morgan’s Rule. ~A given

~B given~A o ~B Conjunction~(AV B) De Morgan’s Rule Q.E.D

11. Constructive Dilemma can be used to set up De Morgan’s Rule.(A⊃ ~B) o (C⊃ ~D) givenAVC~BV~D Constructive Dilemma~(B o D) De Morgan’s Rule

12. Addition can be used to set up De Morgan’s Rule.~A given~AV~B Addition~(A o B) De Morgan’s Rule

13. Distribution can be used in two ways to set up Disjunctive Syllogism (A V B) o (AVC) given~AAV (B o C) DistributionB o C Disjunctive Syllogism

A o (B V C) given(A o B) V (A o C) Distribution~(A o B) givenA o C Disjunctive Syllogism

14. Distribution can be used in two ways to set up Simplification.A V (B o C) given(A V B) o (A V C) Distribution(A V B) Simplification

(A o B) V (A o C) given A o (B V C) DistributionA Simplification

15. If inspection of the premises does not reveal how the conclusion should be derived, consider using the rules of replacement to deconstruct the conclusion. (See the example above)16. Material implication can be used to set up Hypothetical Syllogism

~AVB given ~BVC given

A⊃ B Material ImplicationB⊃ C A ⊃ C Hypothetical Syllogism

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17. Exportation can be used to set up Modus Pones.(A o B)⊃C) givenA

A⊃(B⊃C) ExportationA AboveB⊃C) Modus Ponens

18. Exportation can be used to set up Modus Tollens. A⊃(B⊃C) given(A o B)⊃C Exportation~C given~(A⊃B) Modus Ponens

19. Addition can be used to set up Material Implication.A GivenAV~B Addition

~BVA CommutationB⊃ A Material Implication

20. Transposition can be used to set up Hypothetical SyllogismA⊃B given

~C⊃~BB⊃C TranspositionA⊃B aboveA⊃C Hypothetical Syllogism

21. Transposition can be used to set up Constructive Dilemma.Given

(A⊃B) o (C⊃D) given (~B⊃~A) (~D ⊃~C) Transposition

~BV~D given~AV~C Constructive Dilemma

22. Constructive Dilemma can be used to set up Tautology.(A⊃ C) o (B⊃ C) givenAVBCVC Constructive DilemmaC Tautology

23. Material Implication can be used to set up Tautology.A⊃~A given~AV~A Material Implication~A Tautology

24. Material Implication can be used to set up Distribution.A⊃ (Bo C) given~AV (Bo C) Material Implication (~A V B) o (~AVC) Distribution

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